Quantitative fluid approximation in fractional regimes of transport equations with more invariants

We present an extension of results in a previous paper by the first and the last author [PMP, 2022] about macroscopic limits of linear kinetic equations in (potentially) fractional regimes. More precisely, we develop a unified framework inspired by Ellis and Pinsky [J. Math. Pures Appl., 1975] for operators that preserve mass, momentum and energy, and have microscopic equilibrium with heavy tails (typically polynomial). This paper also generalizes one of Hittmeir and Merino [KRM, 2016] in a related framework. The main difficulty, that leads to our main contribution, is the understanding of the spectrum of the generator in the Fourier space, which is significantly complicated by the lack of spectral gap and the fat tails of the equilibrium. Indeed, the scaling of the eigenelements in the suitable macroscopic rescaling is subtle to handle. In particular, our study uncovered an interesting difference in scaling in the fractional regime, where the transversal wave eigenvalues converge faster to zero than the Boussinesq and acoustic wave eigenvalues.